Cherniga R. M.
Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1112–1119
A mathematical model for fluid transport in peritoneal dialysis is constructed. The model is based on a nonlinear system of two-dimensional partial differential equations with corresponding boundary and initial conditions. Using the classical Lie scheme, we establish that the base system of partial differential equations (under some restrictions on coefficients) is invariant under the infinite-dimensional Lie algebra, which enables us to construct families of exact solutions. Moreover, exact solutions with a more general structure are found using another (non-Lie) technique. Finally, it is shown that some of the solutions obtained describe the hydrostatic pressure and the glucose concentration in peritoneal dialysis.
Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1395-1404
We present a complete description of Lie symmetries for the nonlinear diffusive Lotka-Volterra system. The results are used for the construction of exact solutions of the Lotka-Volterra system, which, in turn, are used for solving the corresponding nonlinear boundary-value problems with zero Neumann conditions. The analytic results are compared with the results of computation based on the finite-element method. We conclude that the obtained exact solutions play an important role in solving Neumann boundary-value problems for the Lotka-Volterra system.
Lie Symmetries, Q-Conditional Symmetries, and Exact Solutions of Nonlinear Systems of Diffusion-Convection Equations
Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1340-1355
A complete description of Lie symmetries is obtained for a class of nonlinear diffusion-convection systems containing two Burgers-type equations with two arbitrary functions. A nonlinear diffusion-convection system with unique symmetry properties that is simultaneously invariant with respect to the generalized Galilei algebra and the operators of Q-conditional symmetries with cubic nonlinearities relative to dependent variables is found. For systems of evolution equations, operators of this sort are found for the first time. For the nonlinear system obtained, a system of Lie and non-Lie ansätze is constructed. Exact solutions, which can be used in solving relevant boundary-value problems, are also found.
Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1409-1421
The classical Lie approach and the method of additional generating conditions are applied to constructing multiparameter families of exact solutions of the generalized Fisher equation, which is a simplification of the known coupled reaction–diffusion system describing spatial segregation of interacting species. The exact solutions are applied to solving nonlinear boundary-value problems with zero Neumann conditions. A comparison of the analytic results and the corresponding numerical calculations shows the importance of the exact solutions obtained for the solution of the generalized Fisher equation.
On new exact solutions of a nonlinear diffusion system that describes the growth of protein crystals
Ukr. Mat. Zh. - 1998. - 50, № 8. - pp. 1106–1120
By using the method of additional generating conditions, we construct multiparameter families of exact solutions of a nonlinear diffusion system that describes the growth of protein crystals. We demonstrate the efficiency of the application of the solutions obtained to the solution of the corresponding nonlinear problem with moving boundary.
Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1262–1270
The Lie symmetries of nonlinear diffusion equations with convection term are completely described. The Lie ansatzes and exact solutions of a certain nonlinear generalization of the Murray equation are constructed. An example of the family of non-Lie solutions of the Murray equation is given.
Application of one constructive method for the construction of non-Lie solutions of nonlinear evolution equations
Ukr. Mat. Zh. - 1997. - 49, № 6. - pp. 814–827
We propose a constructive method for the construction of exact solutions of nonlinear partial differential equations. The method is based on the investigation of a fixed nonlinear partial differential equation (system of partial differential equations) together with an additional condition in the form of a linear ordinary differential equation of higher order. By using this method, we obtain new solutions for nonlinear generalizations of the Fisher equation and for some nonlinear evolution systems that describe real processes in physics, biology, and chemistry.
Ukr. Mat. Zh. - 1996. - 48, № 9. - pp. 1265–1277
For the nonlinear system of partial differential equations, which describes the evolution of temperature and density in TOKAMAK plasmas, multiparameter families of exact solutions are constructed. The solutions are constructed by the Lie-method reduction of initial systems of equations to a system of ordinary differential equations. Examples of non-Lie ansätze and exact solutions are also presented.
Ukr. Mat. Zh. - 1989. - 41, № 12. - pp. 1687–1694
Ukr. Mat. Zh. - 1989. - 41, № 10. - pp. 1349–1357